p-group, metabelian, nilpotent (class 2), monomial, rational
Aliases: C42⋊10Q8, C42.197D4, C23.560C24, C22.2502- 1+4, C22.3342+ 1+4, C4.9(C4⋊Q8), C42⋊9C4.36C2, C42⋊8C4.40C2, C42⋊4C4.28C2, (C22×C4).859C23, (C2×C42).624C22, C22.372(C22×D4), C22.138(C22×Q8), (C22×Q8).167C22, C2.48(C22.29C24), C23.81C23.29C2, C23.78C23.16C2, C2.C42.274C22, C2.25(C23.41C23), C2.48(C23.38C23), C2.20(C2×C4⋊Q8), (C2×C4⋊Q8).37C2, (C2×C4).406(C2×D4), (C2×C4).135(C2×Q8), (C2×C4⋊C4).383C22, (C2×C42.C2).25C2, SmallGroup(128,1392)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C42⋊10Q8
G = < a,b,c,d | a4=b4=c4=1, d2=c2, ab=ba, cac-1=a-1b2, dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=c-1 >
Subgroups: 388 in 224 conjugacy classes, 116 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, Q8, C23, C42, C4⋊C4, C22×C4, C22×C4, C2×Q8, C2.C42, C2×C42, C2×C42, C2×C4⋊C4, C2×C4⋊C4, C42.C2, C4⋊Q8, C22×Q8, C42⋊4C4, C42⋊8C4, C42⋊9C4, C23.78C23, C23.81C23, C2×C42.C2, C2×C4⋊Q8, C42⋊10Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C24, C4⋊Q8, C22×D4, C22×Q8, 2+ 1+4, 2- 1+4, C2×C4⋊Q8, C22.29C24, C23.38C23, C23.41C23, C42⋊10Q8
►Regular action on 128 points
Generators in S128
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)(65 66 67 68)(69 70 71 72)(73 74 75 76)(77 78 79 80)(81 82 83 84)(85 86 87 88)(89 90 91 92)(93 94 95 96)(97 98 99 100)(101 102 103 104)(105 106 107 108)(109 110 111 112)(113 114 115 116)(117 118 119 120)(121 122 123 124)(125 126 127 128)
(1 124 36 30)(2 121 33 31)(3 122 34 32)(4 123 35 29)(5 41 104 72)(6 42 101 69)(7 43 102 70)(8 44 103 71)(9 55 95 87)(10 56 96 88)(11 53 93 85)(12 54 94 86)(13 83 107 51)(14 84 108 52)(15 81 105 49)(16 82 106 50)(17 78 111 63)(18 79 112 64)(19 80 109 61)(20 77 110 62)(21 59 115 91)(22 60 116 92)(23 57 113 89)(24 58 114 90)(25 67 119 38)(26 68 120 39)(27 65 117 40)(28 66 118 37)(45 126 74 100)(46 127 75 97)(47 128 76 98)(48 125 73 99)
(1 125 117 43)(2 98 118 69)(3 127 119 41)(4 100 120 71)(5 122 46 38)(6 31 47 66)(7 124 48 40)(8 29 45 68)(9 109 50 92)(10 18 51 59)(11 111 52 90)(12 20 49 57)(13 21 56 64)(14 114 53 78)(15 23 54 62)(16 116 55 80)(17 84 58 93)(19 82 60 95)(22 87 61 106)(24 85 63 108)(25 72 34 97)(26 44 35 126)(27 70 36 99)(28 42 33 128)(30 73 65 102)(32 75 67 104)(37 101 121 76)(39 103 123 74)(77 105 113 86)(79 107 115 88)(81 89 94 110)(83 91 96 112)
(1 49 117 12)(2 82 118 95)(3 51 119 10)(4 84 120 93)(5 21 46 64)(6 116 47 80)(7 23 48 62)(8 114 45 78)(9 33 50 28)(11 35 52 26)(13 38 56 122)(14 68 53 29)(15 40 54 124)(16 66 55 31)(17 71 58 100)(18 41 59 127)(19 69 60 98)(20 43 57 125)(22 76 61 101)(24 74 63 103)(25 96 34 83)(27 94 36 81)(30 105 65 86)(32 107 67 88)(37 87 121 106)(39 85 123 108)(42 92 128 109)(44 90 126 111)(70 89 99 110)(72 91 97 112)(73 77 102 113)(75 79 104 115)
G:=sub<Sym(128)| (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64)(65,66,67,68)(69,70,71,72)(73,74,75,76)(77,78,79,80)(81,82,83,84)(85,86,87,88)(89,90,91,92)(93,94,95,96)(97,98,99,100)(101,102,103,104)(105,106,107,108)(109,110,111,112)(113,114,115,116)(117,118,119,120)(121,122,123,124)(125,126,127,128), (1,124,36,30)(2,121,33,31)(3,122,34,32)(4,123,35,29)(5,41,104,72)(6,42,101,69)(7,43,102,70)(8,44,103,71)(9,55,95,87)(10,56,96,88)(11,53,93,85)(12,54,94,86)(13,83,107,51)(14,84,108,52)(15,81,105,49)(16,82,106,50)(17,78,111,63)(18,79,112,64)(19,80,109,61)(20,77,110,62)(21,59,115,91)(22,60,116,92)(23,57,113,89)(24,58,114,90)(25,67,119,38)(26,68,120,39)(27,65,117,40)(28,66,118,37)(45,126,74,100)(46,127,75,97)(47,128,76,98)(48,125,73,99), (1,125,117,43)(2,98,118,69)(3,127,119,41)(4,100,120,71)(5,122,46,38)(6,31,47,66)(7,124,48,40)(8,29,45,68)(9,109,50,92)(10,18,51,59)(11,111,52,90)(12,20,49,57)(13,21,56,64)(14,114,53,78)(15,23,54,62)(16,116,55,80)(17,84,58,93)(19,82,60,95)(22,87,61,106)(24,85,63,108)(25,72,34,97)(26,44,35,126)(27,70,36,99)(28,42,33,128)(30,73,65,102)(32,75,67,104)(37,101,121,76)(39,103,123,74)(77,105,113,86)(79,107,115,88)(81,89,94,110)(83,91,96,112), (1,49,117,12)(2,82,118,95)(3,51,119,10)(4,84,120,93)(5,21,46,64)(6,116,47,80)(7,23,48,62)(8,114,45,78)(9,33,50,28)(11,35,52,26)(13,38,56,122)(14,68,53,29)(15,40,54,124)(16,66,55,31)(17,71,58,100)(18,41,59,127)(19,69,60,98)(20,43,57,125)(22,76,61,101)(24,74,63,103)(25,96,34,83)(27,94,36,81)(30,105,65,86)(32,107,67,88)(37,87,121,106)(39,85,123,108)(42,92,128,109)(44,90,126,111)(70,89,99,110)(72,91,97,112)(73,77,102,113)(75,79,104,115)>;
G:=Group( (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64)(65,66,67,68)(69,70,71,72)(73,74,75,76)(77,78,79,80)(81,82,83,84)(85,86,87,88)(89,90,91,92)(93,94,95,96)(97,98,99,100)(101,102,103,104)(105,106,107,108)(109,110,111,112)(113,114,115,116)(117,118,119,120)(121,122,123,124)(125,126,127,128), (1,124,36,30)(2,121,33,31)(3,122,34,32)(4,123,35,29)(5,41,104,72)(6,42,101,69)(7,43,102,70)(8,44,103,71)(9,55,95,87)(10,56,96,88)(11,53,93,85)(12,54,94,86)(13,83,107,51)(14,84,108,52)(15,81,105,49)(16,82,106,50)(17,78,111,63)(18,79,112,64)(19,80,109,61)(20,77,110,62)(21,59,115,91)(22,60,116,92)(23,57,113,89)(24,58,114,90)(25,67,119,38)(26,68,120,39)(27,65,117,40)(28,66,118,37)(45,126,74,100)(46,127,75,97)(47,128,76,98)(48,125,73,99), (1,125,117,43)(2,98,118,69)(3,127,119,41)(4,100,120,71)(5,122,46,38)(6,31,47,66)(7,124,48,40)(8,29,45,68)(9,109,50,92)(10,18,51,59)(11,111,52,90)(12,20,49,57)(13,21,56,64)(14,114,53,78)(15,23,54,62)(16,116,55,80)(17,84,58,93)(19,82,60,95)(22,87,61,106)(24,85,63,108)(25,72,34,97)(26,44,35,126)(27,70,36,99)(28,42,33,128)(30,73,65,102)(32,75,67,104)(37,101,121,76)(39,103,123,74)(77,105,113,86)(79,107,115,88)(81,89,94,110)(83,91,96,112), (1,49,117,12)(2,82,118,95)(3,51,119,10)(4,84,120,93)(5,21,46,64)(6,116,47,80)(7,23,48,62)(8,114,45,78)(9,33,50,28)(11,35,52,26)(13,38,56,122)(14,68,53,29)(15,40,54,124)(16,66,55,31)(17,71,58,100)(18,41,59,127)(19,69,60,98)(20,43,57,125)(22,76,61,101)(24,74,63,103)(25,96,34,83)(27,94,36,81)(30,105,65,86)(32,107,67,88)(37,87,121,106)(39,85,123,108)(42,92,128,109)(44,90,126,111)(70,89,99,110)(72,91,97,112)(73,77,102,113)(75,79,104,115) );
G=PermutationGroup([[(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64),(65,66,67,68),(69,70,71,72),(73,74,75,76),(77,78,79,80),(81,82,83,84),(85,86,87,88),(89,90,91,92),(93,94,95,96),(97,98,99,100),(101,102,103,104),(105,106,107,108),(109,110,111,112),(113,114,115,116),(117,118,119,120),(121,122,123,124),(125,126,127,128)], [(1,124,36,30),(2,121,33,31),(3,122,34,32),(4,123,35,29),(5,41,104,72),(6,42,101,69),(7,43,102,70),(8,44,103,71),(9,55,95,87),(10,56,96,88),(11,53,93,85),(12,54,94,86),(13,83,107,51),(14,84,108,52),(15,81,105,49),(16,82,106,50),(17,78,111,63),(18,79,112,64),(19,80,109,61),(20,77,110,62),(21,59,115,91),(22,60,116,92),(23,57,113,89),(24,58,114,90),(25,67,119,38),(26,68,120,39),(27,65,117,40),(28,66,118,37),(45,126,74,100),(46,127,75,97),(47,128,76,98),(48,125,73,99)], [(1,125,117,43),(2,98,118,69),(3,127,119,41),(4,100,120,71),(5,122,46,38),(6,31,47,66),(7,124,48,40),(8,29,45,68),(9,109,50,92),(10,18,51,59),(11,111,52,90),(12,20,49,57),(13,21,56,64),(14,114,53,78),(15,23,54,62),(16,116,55,80),(17,84,58,93),(19,82,60,95),(22,87,61,106),(24,85,63,108),(25,72,34,97),(26,44,35,126),(27,70,36,99),(28,42,33,128),(30,73,65,102),(32,75,67,104),(37,101,121,76),(39,103,123,74),(77,105,113,86),(79,107,115,88),(81,89,94,110),(83,91,96,112)], [(1,49,117,12),(2,82,118,95),(3,51,119,10),(4,84,120,93),(5,21,46,64),(6,116,47,80),(7,23,48,62),(8,114,45,78),(9,33,50,28),(11,35,52,26),(13,38,56,122),(14,68,53,29),(15,40,54,124),(16,66,55,31),(17,71,58,100),(18,41,59,127),(19,69,60,98),(20,43,57,125),(22,76,61,101),(24,74,63,103),(25,96,34,83),(27,94,36,81),(30,105,65,86),(32,107,67,88),(37,87,121,106),(39,85,123,108),(42,92,128,109),(44,90,126,111),(70,89,99,110),(72,91,97,112),(73,77,102,113),(75,79,104,115)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 4A | 4B | 4C | 4D | 4E | ··· | 4P | 4Q | ··· | 4X |
| order | 1 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | 2+ 1+4 | 2- 1+4 |
| kernel | C42⋊10Q8 | C42⋊4C4 | C42⋊8C4 | C42⋊9C4 | C23.78C23 | C23.81C23 | C2×C42.C2 | C2×C4⋊Q8 | C42 | C42 | C22 | C22 |
| # reps | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 4 | 8 | 2 | 2 |
Matrix representation of C42⋊10Q8 ►in GL8(𝔽5)
| 4 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 3 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 | 1 | 2 |
| 0 | 0 | 0 | 0 | 4 | 3 | 4 | 4 |
| 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 4 | 3 |
| 0 | 0 | 0 | 0 | 0 | 2 | 0 | 1 |
| 1 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 3 |
| 0 | 0 | 0 | 0 | 1 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 4 | 2 | 2 | 0 |
G:=sub<GL(8,GF(5))| [4,4,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,0,0,2,0,1,4,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,1,0,4,0,0,0,0,3,1,2,3,0,0,0,0,0,0,1,4,0,0,0,0,0,0,2,4],[2,2,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,4,3,2,0,0,0,0,0,0,4,0,0,0,0,0,0,0,3,1],[1,1,0,0,0,0,0,0,3,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,2,0,1,4,0,0,0,0,0,0,0,2,0,0,0,0,2,3,3,2,0,0,0,0,0,3,0,0] >;
C42⋊10Q8 in GAP, Magma, Sage, TeX
C_4^2\rtimes_{10}Q_8 % in TeX
G:=Group("C4^2:10Q8"); // GroupNames label
G:=SmallGroup(128,1392);
// by ID
G=gap.SmallGroup(128,1392);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,456,758,723,184,185,80]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=c^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations